Types and coalgebraic structure

We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class Set(F) of all F-coalgebras. In particular, we give coalgebraic characterizations for the condition that F weakly preserves pullbacks, kernel pairs or preimages. We also describe regular monos and epis. In case that vertical bar F(1)vertical bar not equal 1 we show that F preserves preimages iff HS(K) = SH(K) for every class K of F-coalgebras. The case vertical bar F (1)vertical bar = 1 is left as an open problem.